Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.2% → 99.8%

Time: 9.5s

Alternatives: 11

Speedup: 0.6×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * ((y - z) + 1.0d0)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * ((y - z) + 1.0)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 88.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * ((y - z) + 1.0d0)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * ((y - z) + 1.0)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-17} \lor \neg \left(z \leq 8.8 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4e-17) (not (<= z 8.8e-51)))
   (* x (/ (- y (+ z -1.0)) z))
   (/ (+ x (* x y)) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e-17) || !(z <= 8.8e-51)) {
		tmp = x * ((y - (z + -1.0)) / z);
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-4d-17)) .or. (.not. (z <= 8.8d-51))) then
        tmp = x * ((y - (z + (-1.0d0))) / z)
    else
        tmp = (x + (x * y)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4e-17) || !(z <= 8.8e-51)) {
		tmp = x * ((y - (z + -1.0)) / z);
	} else {
		tmp = (x + (x * y)) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4e-17) or not (z <= 8.8e-51):
		tmp = x * ((y - (z + -1.0)) / z)
	else:
		tmp = (x + (x * y)) / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4e-17) || !(z <= 8.8e-51))
		tmp = Float64(x * Float64(Float64(y - Float64(z + -1.0)) / z));
	else
		tmp = Float64(Float64(x + Float64(x * y)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4e-17) || ~((z <= 8.8e-51)))
		tmp = x * ((y - (z + -1.0)) / z);
	else
		tmp = (x + (x * y)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4e-17], N[Not[LessEqual[z, 8.8e-51]], $MachinePrecision]], N[(x * N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.00000000000000029e-17 or 8.8000000000000001e-51 < z

   1. Initial program 82.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]

      2. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) + 1}{z} \cdot x} \]

      3. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{y - \left(z - 1\right)}}{z} \cdot x \]

      4. sub-neg 99.9%

         \[\leadsto \frac{y - \color{blue}{\left(z + \left(-1\right)\right)}}{z} \cdot x \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{y - \left(z + \color{blue}{-1}\right)}{z} \cdot x \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y - \left(z + -1\right)}{z} \cdot x} \]

   if -4.00000000000000029e-17 < z < 8.8000000000000001e-51

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]

      2. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x \cdot 1\right)}}{z} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y - z, \color{blue}{x}\right)}{z} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.9%

      \[\leadsto \color{blue}{\frac{x + x \cdot y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-17} \lor \neg \left(z \leq 8.8 \cdot 10^{-51}\right):\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 10000000.0) (/ (fma x (- y z) x) z) (* x (/ (- y (+ z -1.0)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 10000000.0) {
		tmp = fma(x, (y - z), x) / z;
	} else {
		tmp = x * ((y - (z + -1.0)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 10000000.0)
		tmp = Float64(fma(x, Float64(y - z), x) / z);
	else
		tmp = Float64(x * Float64(Float64(y - Float64(z + -1.0)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 10000000.0], N[(N[(x * N[(y - z), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e7

   1. Initial program 93.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. distribute-lft-in 93.7%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]

      2. fma-define 93.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x \cdot 1\right)}}{z} \]

      3. *-rgt-identity 93.7%

         \[\leadsto \frac{\mathsf{fma}\left(x, y - z, \color{blue}{x}\right)}{z} \]

   3. Simplified 93.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]
   4. Add Preprocessing

   if 1e7 < x

   1. Initial program 83.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]

      2. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) + 1}{z} \cdot x} \]

      3. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{y - \left(z - 1\right)}}{z} \cdot x \]

      4. sub-neg 99.9%

         \[\leadsto \frac{y - \color{blue}{\left(z + \left(-1\right)\right)}}{z} \cdot x \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{y - \left(z + \color{blue}{-1}\right)}{z} \cdot x \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y - \left(z + -1\right)}{z} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 65.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2 \cdot 10^{+55}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-12}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-143}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq -2.4 \cdot 10^{-252}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -4.4 \cdot 10^{-276}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-169}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.65 \cdot 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ x z))))
   (if (<= z -2e+55)
     (- x)
     (if (<= z -1.7e-12)
       (* x (/ y z))
       (if (<= z -1.55e-143)
         (/ x z)
         (if (<= z -2.4e-252)
           t_0
           (if (<= z -4.4e-276)
             (/ x z)
             (if (<= z 2.8e-169)
               t_0
               (if (<= z 5.4e-80)
                 (/ x z)
                 (if (<= z 1.65e+18) t_0 (- x)))))))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -2e+55) {
		tmp = -x;
	} else if (z <= -1.7e-12) {
		tmp = x * (y / z);
	} else if (z <= -1.55e-143) {
		tmp = x / z;
	} else if (z <= -2.4e-252) {
		tmp = t_0;
	} else if (z <= -4.4e-276) {
		tmp = x / z;
	} else if (z <= 2.8e-169) {
		tmp = t_0;
	} else if (z <= 5.4e-80) {
		tmp = x / z;
	} else if (z <= 1.65e+18) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = y * (x / z)
    if (z <= (-2d+55)) then
        tmp = -x
    else if (z <= (-1.7d-12)) then
        tmp = x * (y / z)
    else if (z <= (-1.55d-143)) then
        tmp = x / z
    else if (z <= (-2.4d-252)) then
        tmp = t_0
    else if (z <= (-4.4d-276)) then
        tmp = x / z
    else if (z <= 2.8d-169) then
        tmp = t_0
    else if (z <= 5.4d-80) then
        tmp = x / z
    else if (z <= 1.65d+18) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (x / z);
	double tmp;
	if (z <= -2e+55) {
		tmp = -x;
	} else if (z <= -1.7e-12) {
		tmp = x * (y / z);
	} else if (z <= -1.55e-143) {
		tmp = x / z;
	} else if (z <= -2.4e-252) {
		tmp = t_0;
	} else if (z <= -4.4e-276) {
		tmp = x / z;
	} else if (z <= 2.8e-169) {
		tmp = t_0;
	} else if (z <= 5.4e-80) {
		tmp = x / z;
	} else if (z <= 1.65e+18) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (x / z)
	tmp = 0
	if z <= -2e+55:
		tmp = -x
	elif z <= -1.7e-12:
		tmp = x * (y / z)
	elif z <= -1.55e-143:
		tmp = x / z
	elif z <= -2.4e-252:
		tmp = t_0
	elif z <= -4.4e-276:
		tmp = x / z
	elif z <= 2.8e-169:
		tmp = t_0
	elif z <= 5.4e-80:
		tmp = x / z
	elif z <= 1.65e+18:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (z <= -2e+55)
		tmp = Float64(-x);
	elseif (z <= -1.7e-12)
		tmp = Float64(x * Float64(y / z));
	elseif (z <= -1.55e-143)
		tmp = Float64(x / z);
	elseif (z <= -2.4e-252)
		tmp = t_0;
	elseif (z <= -4.4e-276)
		tmp = Float64(x / z);
	elseif (z <= 2.8e-169)
		tmp = t_0;
	elseif (z <= 5.4e-80)
		tmp = Float64(x / z);
	elseif (z <= 1.65e+18)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (x / z);
	tmp = 0.0;
	if (z <= -2e+55)
		tmp = -x;
	elseif (z <= -1.7e-12)
		tmp = x * (y / z);
	elseif (z <= -1.55e-143)
		tmp = x / z;
	elseif (z <= -2.4e-252)
		tmp = t_0;
	elseif (z <= -4.4e-276)
		tmp = x / z;
	elseif (z <= 2.8e-169)
		tmp = t_0;
	elseif (z <= 5.4e-80)
		tmp = x / z;
	elseif (z <= 1.65e+18)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e+55], (-x), If[LessEqual[z, -1.7e-12], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.55e-143], N[(x / z), $MachinePrecision], If[LessEqual[z, -2.4e-252], t$95$0, If[LessEqual[z, -4.4e-276], N[(x / z), $MachinePrecision], If[LessEqual[z, 2.8e-169], t$95$0, If[LessEqual[z, 5.4e-80], N[(x / z), $MachinePrecision], If[LessEqual[z, 1.65e+18], t$95$0, (-x)]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.00000000000000002e55 or 1.65e18 < z

   1. Initial program 78.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 78.3%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg 78.3%

         \[\leadsto \color{blue}{-x} \]

   5. Simplified 78.3%

      \[\leadsto \color{blue}{-x} \]

   if -2.00000000000000002e55 < z < -1.7e-12

   1. Initial program 99.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 62.9%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-/l* 63.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   5. Simplified 63.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   if -1.7e-12 < z < -1.55000000000000004e-143 or -2.4000000000000002e-252 < z < -4.39999999999999961e-276 or 2.79999999999999988e-169 < z < 5.4000000000000004e-80

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]

      2. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x \cdot 1\right)}}{z} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y - z, \color{blue}{x}\right)}{z} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.8%

      \[\leadsto \color{blue}{\frac{x + x \cdot y}{z}} \]

   6. Taylor expanded in y around 0 74.8%

      \[\leadsto \color{blue}{\frac{x}{z}} \]

   if -1.55000000000000004e-143 < z < -2.4000000000000002e-252 or -4.39999999999999961e-276 < z < 2.79999999999999988e-169 or 5.4000000000000004e-80 < z < 1.65e18

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 74.5%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. *-commutative 74.5%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{z} \]

      2. associate-/l* 80.9%

         \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

   5. Applied egg-rr 80.9%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 4: 62.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+55}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-277}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-166}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;z \leq 10^{+18}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ y z))))
   (if (<= z -1.9e+55)
     (- x)
     (if (<= z -3.3e-12)
       t_0
       (if (<= z -2.6e-277)
         (/ x z)
         (if (<= z 1.95e-166)
           t_0
           (if (<= z 2.5e-75) (/ x z) (if (<= z 1e+18) t_0 (- x)))))))))

C

double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (z <= -1.9e+55) {
		tmp = -x;
	} else if (z <= -3.3e-12) {
		tmp = t_0;
	} else if (z <= -2.6e-277) {
		tmp = x / z;
	} else if (z <= 1.95e-166) {
		tmp = t_0;
	} else if (z <= 2.5e-75) {
		tmp = x / z;
	} else if (z <= 1e+18) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (y / z)
    if (z <= (-1.9d+55)) then
        tmp = -x
    else if (z <= (-3.3d-12)) then
        tmp = t_0
    else if (z <= (-2.6d-277)) then
        tmp = x / z
    else if (z <= 1.95d-166) then
        tmp = t_0
    else if (z <= 2.5d-75) then
        tmp = x / z
    else if (z <= 1d+18) then
        tmp = t_0
    else
        tmp = -x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (y / z);
	double tmp;
	if (z <= -1.9e+55) {
		tmp = -x;
	} else if (z <= -3.3e-12) {
		tmp = t_0;
	} else if (z <= -2.6e-277) {
		tmp = x / z;
	} else if (z <= 1.95e-166) {
		tmp = t_0;
	} else if (z <= 2.5e-75) {
		tmp = x / z;
	} else if (z <= 1e+18) {
		tmp = t_0;
	} else {
		tmp = -x;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (y / z)
	tmp = 0
	if z <= -1.9e+55:
		tmp = -x
	elif z <= -3.3e-12:
		tmp = t_0
	elif z <= -2.6e-277:
		tmp = x / z
	elif z <= 1.95e-166:
		tmp = t_0
	elif z <= 2.5e-75:
		tmp = x / z
	elif z <= 1e+18:
		tmp = t_0
	else:
		tmp = -x
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (z <= -1.9e+55)
		tmp = Float64(-x);
	elseif (z <= -3.3e-12)
		tmp = t_0;
	elseif (z <= -2.6e-277)
		tmp = Float64(x / z);
	elseif (z <= 1.95e-166)
		tmp = t_0;
	elseif (z <= 2.5e-75)
		tmp = Float64(x / z);
	elseif (z <= 1e+18)
		tmp = t_0;
	else
		tmp = Float64(-x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (y / z);
	tmp = 0.0;
	if (z <= -1.9e+55)
		tmp = -x;
	elseif (z <= -3.3e-12)
		tmp = t_0;
	elseif (z <= -2.6e-277)
		tmp = x / z;
	elseif (z <= 1.95e-166)
		tmp = t_0;
	elseif (z <= 2.5e-75)
		tmp = x / z;
	elseif (z <= 1e+18)
		tmp = t_0;
	else
		tmp = -x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.9e+55], (-x), If[LessEqual[z, -3.3e-12], t$95$0, If[LessEqual[z, -2.6e-277], N[(x / z), $MachinePrecision], If[LessEqual[z, 1.95e-166], t$95$0, If[LessEqual[z, 2.5e-75], N[(x / z), $MachinePrecision], If[LessEqual[z, 1e+18], t$95$0, (-x)]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.9e55 or 1e18 < z

   1. Initial program 78.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 78.3%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg 78.3%

         \[\leadsto \color{blue}{-x} \]

   5. Simplified 78.3%

      \[\leadsto \color{blue}{-x} \]

   if -1.9e55 < z < -3.3000000000000001e-12 or -2.6e-277 < z < 1.95e-166 or 2.49999999999999989e-75 < z < 1e18

   1. Initial program 99.8%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 73.3%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. associate-/l* 60.0%

         \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   5. Simplified 60.0%

      \[\leadsto \color{blue}{x \cdot \frac{y}{z}} \]

   if -3.3000000000000001e-12 < z < -2.6e-277 or 1.95e-166 < z < 2.49999999999999989e-75

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]

      2. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x \cdot 1\right)}}{z} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y - z, \color{blue}{x}\right)}{z} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.8%

      \[\leadsto \color{blue}{\frac{x + x \cdot y}{z}} \]

   6. Taylor expanded in y around 0 67.3%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 92.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.82 \cdot 10^{+22} \lor \neg \left(y \leq 500000000000\right):\\ \;\;\;\;\left(y - \left(z + -1\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.82e+22) (not (<= y 500000000000.0)))
   (* (- y (+ z -1.0)) (/ x z))
   (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.82e+22) || !(y <= 500000000000.0)) {
		tmp = (y - (z + -1.0)) * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.82d+22)) .or. (.not. (y <= 500000000000.0d0))) then
        tmp = (y - (z + (-1.0d0))) * (x / z)
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.82e+22) || !(y <= 500000000000.0)) {
		tmp = (y - (z + -1.0)) * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.82e+22) or not (y <= 500000000000.0):
		tmp = (y - (z + -1.0)) * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.82e+22) || !(y <= 500000000000.0))
		tmp = Float64(Float64(y - Float64(z + -1.0)) * Float64(x / z));
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.82e+22) || ~((y <= 500000000000.0)))
		tmp = (y - (z + -1.0)) * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.82e+22], N[Not[LessEqual[y, 500000000000.0]], $MachinePrecision]], N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.82e22 or 5e11 < y

   1. Initial program 91.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative 91.2%

         \[\leadsto \frac{\color{blue}{\left(\left(y - z\right) + 1\right) \cdot x}}{z} \]

      2. associate-/l* 91.7%

         \[\leadsto \color{blue}{\left(\left(y - z\right) + 1\right) \cdot \frac{x}{z}} \]

      3. associate-+l- 91.7%

         \[\leadsto \color{blue}{\left(y - \left(z - 1\right)\right)} \cdot \frac{x}{z} \]

      4. sub-neg 91.7%

         \[\leadsto \left(y - \color{blue}{\left(z + \left(-1\right)\right)}\right) \cdot \frac{x}{z} \]

      5. metadata-eval 91.7%

         \[\leadsto \left(y - \left(z + \color{blue}{-1}\right)\right) \cdot \frac{x}{z} \]

   4. Applied egg-rr 91.7%

      \[\leadsto \color{blue}{\left(y - \left(z + -1\right)\right) \cdot \frac{x}{z}} \]

   if -1.82e22 < y < 5e11

   1. Initial program 91.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 90.0%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 - z\right)}}{z} \]
   4. Step-by-step derivation

      1. sub-neg 90.0%

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \left(-z\right)\right)}}{z} \]

      2. distribute-lft-in 90.0%

         \[\leadsto \frac{\color{blue}{x \cdot 1 + x \cdot \left(-z\right)}}{z} \]

      3. *-rgt-identity 90.0%

         \[\leadsto \frac{\color{blue}{x} + x \cdot \left(-z\right)}{z} \]

      4. distribute-rgt-neg-in 90.0%

         \[\leadsto \frac{x + \color{blue}{\left(-x \cdot z\right)}}{z} \]

      5. unsub-neg 90.0%

         \[\leadsto \frac{\color{blue}{x - x \cdot z}}{z} \]

   5. Simplified 90.0%

      \[\leadsto \frac{\color{blue}{x - x \cdot z}}{z} \]

   6. Taylor expanded in z around inf 98.3%

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x}{z}} \]
   7. Step-by-step derivation

      1. mul-1-neg 98.3%

         \[\leadsto \color{blue}{\left(-x\right)} + \frac{x}{z} \]

      2. +-commutative 98.3%

         \[\leadsto \color{blue}{\frac{x}{z} + \left(-x\right)} \]

      3. sub-neg 98.3%

         \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   8. Simplified 98.3%

      \[\leadsto \color{blue}{\frac{x}{z} - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.82 \cdot 10^{+22} \lor \neg \left(y \leq 500000000000\right):\\ \;\;\;\;\left(y - \left(z + -1\right)\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+50} \lor \neg \left(y \leq 1.7 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.2e+50) (not (<= y 1.7e+15))) (/ (* x y) z) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.2e+50) || !(y <= 1.7e+15)) {
		tmp = (x * y) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.2d+50)) .or. (.not. (y <= 1.7d+15))) then
        tmp = (x * y) / z
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.2e+50) || !(y <= 1.7e+15)) {
		tmp = (x * y) / z;
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.2e+50) or not (y <= 1.7e+15):
		tmp = (x * y) / z
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.2e+50) || !(y <= 1.7e+15))
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.2e+50) || ~((y <= 1.7e+15)))
		tmp = (x * y) / z;
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.2e+50], N[Not[LessEqual[y, 1.7e+15]], $MachinePrecision]], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.19999999999999983e50 or 1.7e15 < y

   1. Initial program 92.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 85.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

   if -3.19999999999999983e50 < y < 1.7e15

   1. Initial program 90.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 88.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 - z\right)}}{z} \]
   4. Step-by-step derivation

      1. sub-neg 88.2%

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \left(-z\right)\right)}}{z} \]

      2. distribute-lft-in 88.2%

         \[\leadsto \frac{\color{blue}{x \cdot 1 + x \cdot \left(-z\right)}}{z} \]

      3. *-rgt-identity 88.2%

         \[\leadsto \frac{\color{blue}{x} + x \cdot \left(-z\right)}{z} \]

      4. distribute-rgt-neg-in 88.2%

         \[\leadsto \frac{x + \color{blue}{\left(-x \cdot z\right)}}{z} \]

      5. unsub-neg 88.2%

         \[\leadsto \frac{\color{blue}{x - x \cdot z}}{z} \]

   5. Simplified 88.2%

      \[\leadsto \frac{\color{blue}{x - x \cdot z}}{z} \]

   6. Taylor expanded in z around inf 97.6%

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x}{z}} \]
   7. Step-by-step derivation

      1. mul-1-neg 97.6%

         \[\leadsto \color{blue}{\left(-x\right)} + \frac{x}{z} \]

      2. +-commutative 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} + \left(-x\right)} \]

      3. sub-neg 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   8. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{x}{z} - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+50} \lor \neg \left(y \leq 1.7 \cdot 10^{+15}\right):\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+50} \lor \neg \left(y \leq 9.5 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.3e+50) (not (<= y 9.5e+15))) (* y (/ x z)) (- (/ x z) x)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e+50) || !(y <= 9.5e+15)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.3d+50)) .or. (.not. (y <= 9.5d+15))) then
        tmp = y * (x / z)
    else
        tmp = (x / z) - x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.3e+50) || !(y <= 9.5e+15)) {
		tmp = y * (x / z);
	} else {
		tmp = (x / z) - x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.3e+50) or not (y <= 9.5e+15):
		tmp = y * (x / z)
	else:
		tmp = (x / z) - x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.3e+50) || !(y <= 9.5e+15))
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(Float64(x / z) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.3e+50) || ~((y <= 9.5e+15)))
		tmp = y * (x / z);
	else
		tmp = (x / z) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.3e+50], N[Not[LessEqual[y, 9.5e+15]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / z), $MachinePrecision] - x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.3e50 or 9.5e15 < y

   1. Initial program 92.6%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 85.1%

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   4. Step-by-step derivation

      1. *-commutative 85.1%

         \[\leadsto \frac{\color{blue}{y \cdot x}}{z} \]

      2. associate-/l* 81.4%

         \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

   5. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

   if -3.3e50 < y < 9.5e15

   1. Initial program 90.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 88.2%

      \[\leadsto \frac{\color{blue}{x \cdot \left(1 - z\right)}}{z} \]
   4. Step-by-step derivation

      1. sub-neg 88.2%

         \[\leadsto \frac{x \cdot \color{blue}{\left(1 + \left(-z\right)\right)}}{z} \]

      2. distribute-lft-in 88.2%

         \[\leadsto \frac{\color{blue}{x \cdot 1 + x \cdot \left(-z\right)}}{z} \]

      3. *-rgt-identity 88.2%

         \[\leadsto \frac{\color{blue}{x} + x \cdot \left(-z\right)}{z} \]

      4. distribute-rgt-neg-in 88.2%

         \[\leadsto \frac{x + \color{blue}{\left(-x \cdot z\right)}}{z} \]

      5. unsub-neg 88.2%

         \[\leadsto \frac{\color{blue}{x - x \cdot z}}{z} \]

   5. Simplified 88.2%

      \[\leadsto \frac{\color{blue}{x - x \cdot z}}{z} \]

   6. Taylor expanded in z around inf 97.6%

      \[\leadsto \color{blue}{-1 \cdot x + \frac{x}{z}} \]
   7. Step-by-step derivation

      1. mul-1-neg 97.6%

         \[\leadsto \color{blue}{\left(-x\right)} + \frac{x}{z} \]

      2. +-commutative 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} + \left(-x\right)} \]

      3. sub-neg 97.6%

         \[\leadsto \color{blue}{\frac{x}{z} - x} \]

   8. Simplified 97.6%

      \[\leadsto \color{blue}{\frac{x}{z} - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.3 \cdot 10^{+50} \lor \neg \left(y \leq 9.5 \cdot 10^{+15}\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9200000:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x 9200000.0)
   (/ (* x (+ (- y z) 1.0)) z)
   (* x (/ (- y (+ z -1.0)) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= 9200000.0) {
		tmp = (x * ((y - z) + 1.0)) / z;
	} else {
		tmp = x * ((y - (z + -1.0)) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= 9200000.0d0) then
        tmp = (x * ((y - z) + 1.0d0)) / z
    else
        tmp = x * ((y - (z + (-1.0d0))) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= 9200000.0) {
		tmp = (x * ((y - z) + 1.0)) / z;
	} else {
		tmp = x * ((y - (z + -1.0)) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= 9200000.0:
		tmp = (x * ((y - z) + 1.0)) / z
	else:
		tmp = x * ((y - (z + -1.0)) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= 9200000.0)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z);
	else
		tmp = Float64(x * Float64(Float64(y - Float64(z + -1.0)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= 9200000.0)
		tmp = (x * ((y - z) + 1.0)) / z;
	else
		tmp = x * ((y - (z + -1.0)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, 9200000.0], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(x * N[(N[(y - N[(z + -1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.2e6

   1. Initial program 93.7%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   if 9.2e6 < x

   1. Initial program 83.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto \color{blue}{x \cdot \frac{\left(y - z\right) + 1}{z}} \]

      2. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\left(y - z\right) + 1}{z} \cdot x} \]

      3. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{y - \left(z - 1\right)}}{z} \cdot x \]

      4. sub-neg 99.9%

         \[\leadsto \frac{y - \color{blue}{\left(z + \left(-1\right)\right)}}{z} \cdot x \]

      5. metadata-eval 99.9%

         \[\leadsto \frac{y - \left(z + \color{blue}{-1}\right)}{z} \cdot x \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{y - \left(z + -1\right)}{z} \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9200000:\\ \;\;\;\;\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y - \left(z + -1\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.26 \cdot 10^{-5}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.26e-5))) (- x) (/ x z)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.26e-5)) {
		tmp = -x;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.26d-5))) then
        tmp = -x
    else
        tmp = x / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.26e-5)) {
		tmp = -x;
	} else {
		tmp = x / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.26e-5):
		tmp = -x
	else:
		tmp = x / z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 1.26e-5))
		tmp = Float64(-x);
	else
		tmp = Float64(x / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 1.26e-5)))
		tmp = -x;
	else
		tmp = x / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.26e-5]], $MachinePrecision]], (-x), N[(x / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1.25999999999999996e-5 < z

   1. Initial program 81.3%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 72.9%

      \[\leadsto \color{blue}{-1 \cdot x} \]
   4. Step-by-step derivation

      1. mul-1-neg 72.9%

         \[\leadsto \color{blue}{-x} \]

   5. Simplified 72.9%

      \[\leadsto \color{blue}{-x} \]

   if -1 < z < 1.25999999999999996e-5

   1. Initial program 99.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Step-by-step derivation

      1. distribute-lft-in 99.9%

         \[\leadsto \frac{\color{blue}{x \cdot \left(y - z\right) + x \cdot 1}}{z} \]

      2. fma-define 99.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x, y - z, x \cdot 1\right)}}{z} \]

      3. *-rgt-identity 99.9%

         \[\leadsto \frac{\mathsf{fma}\left(x, y - z, \color{blue}{x}\right)}{z} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, y - z, x\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 99.3%

      \[\leadsto \color{blue}{\frac{x + x \cdot y}{z}} \]

   6. Taylor expanded in y around 0 56.9%

      \[\leadsto \color{blue}{\frac{x}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 1.26 \cdot 10^{-5}\right):\\ \;\;\;\;-x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.0% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x))

C

double code(double x, double y, double z) {
	return -x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -x
end function

Java

public static double code(double x, double y, double z) {
	return -x;
}

Python

def code(x, y, z):
	return -x

Julia

function code(x, y, z)
	return Float64(-x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -x;
end

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 91.4%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 35.0%

   \[\leadsto \color{blue}{-1 \cdot x} \]
4. Step-by-step derivation

   1. mul-1-neg 35.0%

      \[\leadsto \color{blue}{-x} \]

5. Simplified 35.0%

   \[\leadsto \color{blue}{-x} \]
6. Add Preprocessing

Alternative 11: 3.0% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 x)

C

double code(double x, double y, double z) {
	return x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x
end function

Java

public static double code(double x, double y, double z) {
	return x;
}

Python

def code(x, y, z):
	return x

Julia

function code(x, y, z)
	return x
end

MATLAB

function tmp = code(x, y, z)
	tmp = x;
end

Wolfram

code[x_, y_, z_] := x

Derivation

1. Initial program 91.4%

   \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf 27.6%

   \[\leadsto \frac{\color{blue}{-1 \cdot \left(x \cdot z\right)}}{z} \]
4. Step-by-step derivation

   1. mul-1-neg 27.6%

      \[\leadsto \frac{\color{blue}{-x \cdot z}}{z} \]

   2. distribute-rgt-neg-in 27.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(-z\right)}}{z} \]

5. Simplified 27.6%

   \[\leadsto \frac{\color{blue}{x \cdot \left(-z\right)}}{z} \]
6. Step-by-step derivation

   1. *-commutative 27.6%

      \[\leadsto \frac{\color{blue}{\left(-z\right) \cdot x}}{z} \]

   2. associate-/l* 34.4%

      \[\leadsto \color{blue}{\left(-z\right) \cdot \frac{x}{z}} \]

   3. add-sqr-sqrt 20.3%

      \[\leadsto \color{blue}{\left(\sqrt{-z} \cdot \sqrt{-z}\right)} \cdot \frac{x}{z} \]

   4. sqrt-unprod 13.0%

      \[\leadsto \color{blue}{\sqrt{\left(-z\right) \cdot \left(-z\right)}} \cdot \frac{x}{z} \]

   5. sqr-neg 13.0%

      \[\leadsto \sqrt{\color{blue}{z \cdot z}} \cdot \frac{x}{z} \]

   6. sqrt-unprod 8.7%

      \[\leadsto \color{blue}{\left(\sqrt{z} \cdot \sqrt{z}\right)} \cdot \frac{x}{z} \]

   7. add-sqr-sqrt 10.8%

      \[\leadsto \color{blue}{z} \cdot \frac{x}{z} \]

7. Applied egg-rr 10.8%

   \[\leadsto \color{blue}{z \cdot \frac{x}{z}} \]
8. Step-by-step derivation

   1. associate-*r/ 2.9%

      \[\leadsto \color{blue}{\frac{z \cdot x}{z}} \]

   2. associate-*l/ 2.9%

      \[\leadsto \color{blue}{\frac{z}{z} \cdot x} \]

   3. *-inverses 2.9%

      \[\leadsto \color{blue}{1} \cdot x \]

   4. *-lft-identity 2.9%

      \[\leadsto \color{blue}{x} \]

9. Simplified 2.9%

   \[\leadsto \color{blue}{x} \]
10. Add Preprocessing

Developer target: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024097 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (if (< x -2.71483106713436e-162) (- (* (+ 1.0 y) (/ x z)) x) (if (< x 3.874108816439546e-197) (* (* x (+ (- y z) 1.0)) (/ 1.0 z)) (- (* (+ 1.0 y) (/ x z)) x)))

  (/ (* x (+ (- y z) 1.0)) z))